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Probability theory is based on the
mathematics of sets. A set can be
used to define a “Sample Space” which enumerates the possible outcomes of
an “experiment”. Probability can be
defined as the ratio of the number of positive outcomes to the number of
all possible outcomes in an experiment. |
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We will now review some set theory to have
some tools we can use in this course. |
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Later in the course we will also need to
review Sums, Calculus, Integral Transforms, and some Linear Algebra. |
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Set Theory |
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Definitions |
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Laws |
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Functions |
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Set: “A”
is a set of elements {a, b, c, d}
“c” is an element of “A”
Null Set {}
Universal Set “S” |
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Set Algebra
B is contained in A (subset)
A contains B (superset)
A + B (sum, union)
A*B (product, intersection) |
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Mutually Exclusive - Sets that have no elements
in common. A*B = {} |
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Collectively Exhaustive - Sets whose union is
the universal set. A+B+C+D = S |
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Idempotent: A+A=A; A*A=A |
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Commutative A+B=B+A; A*B=B*A |
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Associative A+(B+C)=(A+B)+C
A*(B*C)=(A*B)*C |
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Distributive A*(B+C)=A*B+A*C |
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Product Identities {}*A={}; S*A=A |
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Sum Identities {}+A=A; S+A=S |
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Consistency |
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Universal Bounds |
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Involution |
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Complementary |
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De Morgan’s First Law |
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Demorgan’s Second Law |
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Set Theory |
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Definitions |
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Laws |
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Functions |
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Function - A rule that maps every element, x, of
a given set, D (the domain), into elements, f(x), of another set, R (the
range). |
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Notes